Simpler, more efficient, functional linear maps

A previous post described a data type of functional linear maps. As Andy Gill pointed out, we had a heck of a time trying to get good performance. This note describes a new representation that is very simple and much more efficient. It’s terribly obvious in retrospect but took me a good while to stumble onto.

The Haskell module described here is part of the vector-space library (version 0.5 or later) and requires ghc version 6.10 or better (for associated types).


  • 2008-11-09: Changed remarks about versions. The vector-space version 0.5 depends on ghc 6.10.
  • 2008-10-21: Fixed the vector-space library link in the teaser.

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Vector space bases via type families

A basis B of a vector space V is a subset B of V, such that the elements of B are linearly independent and span V. That is to say, every element (vector) of V is a linear combination of elements of B, and no element of B is a linear combination of the other elements of B. Moreover, every basis determines a unique decomposition of any member of V into coordinates relative to B.

This post gives a simple Haskell implementation for a canonical basis of a vector space, and a means of decomposing vectors into coordinates. It uses [indexed type families] (associated types), and is quite general, despite its simplicity.

The Haskell module described here is part of the vector-space library (version 0.4 or later), which available on Hackage and a darcs repository. See the wiki page, interface documentation, and source code. The library version described below (0.5 or later) relies on ghc 6.10.


  • 2008-11-09: Tweaked comment above about version.
  • 2008-02-09: just fiddling around

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Functional linear maps

Two earlier posts described a simple and general notion of derivative that unifies the many concrete notions taught in traditional calculus courses. All of those variations turn out to be concrete representations of the single abstract notion of a linear map. Correspondingly, the various forms of mulitplication in chain rules all turn out to be implementations of composition of linear maps. For simplicity, I suggested a direct implementation of linear maps as functions. Unfortunately, that direct representation thwarts efficiency, since functions, unlike data structures, do not cache by default.

This post presents a data representation of linear maps that makes crucial use of (a) linearity and (b) the recently added language feature indexed type families (“associated types”).

For a while now, I’ve wondered if a library for linear maps could replace and generalize matrix libraries. After all, matrices represent of a restricted class of linear maps. Unlike conventional matrix libraries, however, the linear map library described in this post captures matrix/linear-map dimensions via static typing. The composition function defined below statically enforces the conformability property required of matrix multiplication (which implements linear map composition). Likewise, conformance for addition of linear maps is also enforced simply and statically. Moreover, with sufficiently sophisticated coaxing of the Haskell compiler, of the sort Don Stewart does, perhaps a library like this one could also have terrific performance. (It doesn’t yet.)

You can read and try out the code for this post in the module Data.LinearMap in version 0.2.0 or later of the vector-space package. That module also contains an implementation of linear map composition, as well as Functor-like and Applicative-like operations. Andy Gill has been helping me get to the bottom of some some severe performance problems, apparently involving huge amounts of redundant dictionary creation.


  • 2008-06-04: Brief explanation of the associated data type declaration.

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